What is the fundamental range equation

The fundamental radar range equation is a crucial formula used to determine the maximum range at which a radar system can detect a target. It relates the power received by the radar to the transmitted power, the characteristics of the radar system, and the properties of the target and environment. The radar range equation is fundamental to understanding radar performance and designing radar systems.

Radar Range Equation

The most basic form of the radar range equation for a monostatic radar system (where the transmitter and receiver are co-located) is given by:

Pr=PtGtGrλ2σ(4π)3R4LP_r = \frac{P_t G_t G_r \lambda^2 \sigma}{(4\pi)^3 R^4 L}Pr?=(4π)3R4LPt?Gt?Gr?λ2σ?

Where:

• PrP_rPr?: Received power
• PtP_tPt?: Transmitted power
• GtG_tGt?: Gain of the transmitting antenna
• GrG_rGr?: Gain of the receiving antenna (often Gr=GtG_r = G_tGr?=Gt? for monostatic radar)
• λ\lambdaλ: Wavelength of the transmitted signal
• σ\sigmaσ: Radar cross-section (RCS) of the target
• RRR: Range to the target
• LLL: System losses (including losses due to propagation, hardware, etc.)

Simplified Equation

For simplicity, assuming a monostatic radar (Gt=GrG_t = G_rGt?=Gr?) and neglecting losses (L≈1L \approx 1L≈1), the equation simplifies to:

Pr=PtG2λ2σ(4π)3R4P_r = \frac{P_t G^2 \lambda^2 \sigma}{(4\pi)^3 R^4}Pr?=(4π)3R4Pt?G2λ2σ?

Where GGG is the antenna gain, and RRR is the range to the target.

Derivation and Explanation

1. Transmitted Power (PtP_tPt?):

   • The radar transmits a power PtP_tPt? through an antenna with gain GtG_tGt?.

2. Propagation and Reflection:

   • The transmitted signal propagates through space and encounters a target with a radar cross-section σ\sigmaσ, which reflects some of the energy back toward the radar.

3. Received Power (PrP_rPr?):

   • The power density of the signal at the target is PtGt4πR2\frac{P_t G_t}{4\pi R^2}4πR2Pt?Gt??.
   • The power reflected back toward the radar is proportional to the target's RCS, σ\sigmaσ.
   • The reflected power spreads out spherically, and the power density at the radar receiver is PtGtσ(4πR2)4πR2\frac{P_t G_t \sigma}{(4\pi R^2) 4\pi R^2}(4πR2)4πR2Pt?Gt?σ?.

4. Antenna Gain (GrG_rGr?):

   • The receiving antenna with gain GrG_rGr? captures the reflected power, and the received power PrP_rPr? is proportional to the power density at the receiver multiplied by the effective aperture of the receiving antenna, which is related to its gain.

Range Determination

The radar range equation can be rearranged to solve for the maximum range RRR at which a target with a given RCS can be detected:

Rmax=(PtGtGrλ2σ(4π)3PrL)14R_{\text{max}} = \left( \frac{P_t G_t G_r \lambda^2 \sigma}{(4\pi)^3 P_r L} \right)^{\frac{1}{4}}Rmax?=((4π)3Pr?LPt?Gt?Gr?λ2σ?)41?

Key Factors Affecting Radar Range:

1. Transmitted Power (PtP_tPt?): Higher transmitted power increases the maximum detection range.
2. Antenna Gain (GtG_tGt? and GrG_rGr?): Higher gain antennas focus the transmitted and received energy, extending the range.
3. Wavelength (λ\lambdaλ): The wavelength is inversely proportional to frequency (λ=cf\lambda = \frac{c}{f}λ=fc?). Lower frequencies (longer wavelengths) typically result in better penetration through obstacles but may require larger antennas.
4. Radar Cross-Section (σ\sigmaσ): A larger RCS means the target reflects more energy, increasing the detection range.
5. System Losses (LLL): Minimizing losses in the system improves the range.

In summary, the fundamental radar range equation provides a quantitative relationship between the radar's transmitted power, antenna characteristics, target properties, and range. It is a critical tool for designing radar systems and understanding their performance in various scenarios.